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Condensation and thermalization of an easy-plane ferromagnet in a spinor Bose gas

Nature Physics volume 18, pages 1459–1463 (2022)Cite this article

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Abstract

Bose–Einstein condensates are an ideal platform to explore dynamical phenomena emerging in the many-body limit, such as the build-up of long-range coherence, superfluidity or spontaneous symmetry breaking. Here we study the thermalization dynamics of an easy-plane ferromagnet employing a homogeneous one-dimensional spinor Bose gas. We demonstrate the dynamic emergence of effective long-range coherence for the spin field and verify spin-superfluidity by experimentally testing Landau’s criterion. We reveal the structure of one massive and two massless emerging modes—a consequence of explicit and spontaneous symmetry breaking, respectively. Our experiments allow us to observe the thermalization of an easy-plane ferromagnetic Bose gas. The relevant momentum-resolved observables are in agreement with a thermal prediction obtained from an underlying microscopic model within the Bogoliubov approximation. Our methods and results are a step towards a quantitative understanding of condensation dynamics in large magnetic spin systems and the study of the role of entanglement and topological excitations for their thermalization.

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Fig. 1: Homogeneous spinor Bose gas and easy-plane ferromagnetic properties.
Fig. 2: Emergence of effective long-range coherence and superfluidity.
Fig. 3: Local spin and density control enables probing of quasiparticle properties.
Fig. 4: Structure factors of different observables at late times.

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Data availability

Data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. Source data are provided with this paper.

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Acknowledgements

We thank J. Dreher for experimental assistance. This work is supported by ERC Advanced Grant Horizon 2020 EntangleGen (Project-ID 694561), the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC2181/1-390900948 (the Heidelberg STRUCTURES Excellence Cluster) and the Collaborative Research Center, Project-ID 27381115, SFB 1225 ISOQUANT. M.P. has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 101032523.

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Authors and Affiliations

  1. Kirchhoff-Institut für Physik, Universität Heidelberg, Heidelberg, Germany

    Maximilian Prüfer, Stefan Lannig, Helmut Strobel & Markus K. Oberthaler

  2. Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Wien, Austria

    Maximilian Prüfer

  3. Institut für Theoretische Physik, Universität Heidelberg, Heidelberg, Germany

    Daniel Spitz & Jürgen Berges

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Contributions

M.P., S.L. and H.S. took the measurement data. M.P., D.S., S.L., H.S. and M.K.O. discussed the measurement results and analysed the data. D.S., M.P. and J.B. elaborated the theoretical framework. All authors contributed to the discussion of the results and the writing of the manuscript.

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Correspondence to Maximilian Prüfer.

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Extended data

Extended Data Fig. 1 Measurement of the gap by observation of temporal oscillations of the k = 0 mode.

a, We measure the gap of the quadratic spin mode by a global rotation of the spinor phase. We record the resulting oscillations of the fractional m = 0 population as a function of evolution time after the rotation. We fit a sinusoidal function (solid line) to infer the frequency. b, Extracted oscillation frequency (diamonds) and mean value of the m = 0 population (circles). We compare to theoretical expectations for the easy-plane phase (solid lines; see Methods equation 1). The dashed line extrapolates the expectations to q < 0 under the assumption of equal populations of m = ± 1. For the theory curves we use nc1 = 1.3Hz.

Extended Data Fig. 2 Histograms of local observables in the thermalized state.

Histograms obtained from evaluating the local observations of the experimental data presented in Fig. 4 (green bars). Here, each local observable is normalized to the square-root of the local mean of the total atom number. On top we display theoretical estimates from 1000 samples generated according to thermal Bogoliubov theory with parameters as in Fig. 4 (grey line; grey band indicates 68% confidence interval including statistical and systematic uncertainties). The mean value of each histogram is subtracted. For details on the sampling procedure see Methods.

Extended Data Fig. 3 Structure factor close to q = 0.

We show experimental power spectra of different spin and density degrees of freedom close to q = 0 (green diamonds). The grey diamonds represent the fluctuations of a coherent spin state with comparable atom numbers. We compare to thermal Bogoliubov theory predictions for the same parameters as displayed in Fig. 4 but with q = 0 (green line; grey band indicates 68% confidence interval of statistical and systematic uncertainties). Experimentally, we find that for momenta in the range of 0.02 μm−1 to 0.1 μm−1 the fluctuations are higher than for the thermal predictions for all observables (except the transversal spin F⊥). The length scale of these fluctuations is in accordance with observable localized long-lived non-linear excitations which are not present in the thermalized data of Fig. 4.

Source data

Source Data Fig. 4

Source data, experiment (thermal state) and Bogoliubov.

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Prüfer, M., Spitz, D., Lannig, S. et al. Condensation and thermalization of an easy-plane ferromagnet in a spinor Bose gas. Nat. Phys. 18, 1459–1463 (2022). https://doi.org/10.1038/s41567-022-01779-6

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